A specific problem is the complexity of small matrix multiplication or rank of the corresponding tensor. In the case of matrices of the size \(2\times 2\) , the tensor has the size \(4\times 4\times 4\)  and it was proved that the rank is 7 [1]}.
However, for matrices of the size \(3\times 3\) , we only know the bounds for this rank [2]}. The lower bound is 19 and the upper bound is 23.
A lot of effort has been exerted to improve the algorithm of Laderman [3]} (decomposition with rank 23), there are many algorithms with the same complexity [4]}, [5]}, [6]}, [7]}, [8]}, but the
proof that a decomposition to a rank smaller than 23 is impossible is not known yet.
For more introductory texts to the subject see, e.g., [9]}, [3]}, [11]}, [12]}.
Utilization of the fast small matrix multiplication in practice is discussed in [13]}, [14]}.
